A Distributional Framework Based on Gamma–Zeta Operators for Singular Fractional Models

Fractional calculus and distribution theory share a common conceptual origin in the symbolic interpretation of differentiation and integration. Despite this connection, most developments in fractional calculus have traditionally been formulated within the framework of ordinary functions, while the systematic use of distributions remains limited. In this work, a novel distributional framework is developed by constructing a fractional Taylor representation of the product of Euler gamma and Riemann zeta functions in terms of fractional derivatives of the Dirac delta distribution. The proposed formulation enables the derivation of new fractional identities via Laplace transformation and facilitates the analytical solution of fractional differential equations containing such functions. Closed-form solutions are obtained in both classical and generalized distributional senses, allowing the extension of solutions from the positive real axis to the entire real line. Furthermore, the framework is applied to fractional operators of Erdélyi–Kober type, yielding new integral and derivative transforms. Fractional differential and integral equations with singular terms arise naturally in several engineering models involving memory effects, impulsive responses, and anomalous transport phenomena. However, the presence of nonremovable singularities—such as those associated with Euler gamma and Riemann zeta functions—significantly restricts the applicability of classical analytical methods. Overall, the proposed distributional framework bridges the gap between abstract fractional calculus and practical engineering models by enabling analytical solutions of fractional systems with singular memory kernels that were previously inaccessible using classical methods.